Archive for the 'Mathematics' Category

Mayan Number Paper…Revealed!

Ok, so the paper I co-wrote with my professor has finally been published! I didn’t even get any warning. About two weeks ago a packet shows up on my doorstep containing five copies of the magazine Mathematics Teacher with my article in it. The subject of our article was even on the front cover!

From what I can tell, Mathematics Teacher is basically like an industry magazine for math teachers, going over various teaching topics and ideas and the like, which is cool. I never heard about it before but, a publication is a publication.

Unfortunately, my contribution seems to be almost entire cut out of the article entirely. It’s not too surprising given that I only solved on specific problem in a textbook and the article is about teaching Mayan numerals in general, but is still a bit disappointing. At least my name is still on the byline:

So anyway, what was the problem I solved? Well to understand that, let’s delve a little bit into the Mayan number system. Mayan numbers are actually not too different from ours. They are positional, so each digit is a multiple of some base unit like ours. For example, in our system, each number is multiplied by a factor of ten. So 50 is 5 × 10, 3,000 is 3 × 103­, and so on. Each time you move a digit to the left, you’re multiplying it by 10. Same thing in the Mayan system, except that for them, you’re multiplying by twenty. Also, the Mayans wrote their numbers from top to bottom, instead of left to right, but that’s just a small detail.

The Mayans had unique symbols for each digit from 0 to 19 to stick in for each digit (well really, they only had three symbols, the dot (for 1), the line (for 5), and the shell (for 0). Kind of similar to tally marks). I’m not going to use the symbols for this post since it would be too confusing and too much work for me (you can see them here), so I’m going to use the typical notation of A.B.C.D… and so on, where each letter is a digit and the larger digit places are towards the left. So, for a quick example, 1024 in our system would be 2.15.4.

The only really weird thing about the Mayan numeral system is that there is a little “jog” in it. While almost every place is twenty times the value of the previous place, the third place (which would be the 202, or 400s place) is actually only 18 times the previous one. Why? Who knows? Although 18 × 20 is 360 which is very close to the number of days per year, so it’s possible that’s why. In the Mayan Long Count Calendar (responsible for the 12/21/2012 end of the world nonsense), 1.0.0 days is called a tun, about equal to a year, and is obviously a nice round number, rather than 18.0 (or really, 18.5 to be more accurate) if it were truly a pure base-20 system.

So what was the exact problem I solved? Well, given what I said about the numbering system with its strange twist, is there an easy and quick way to multiply any given Mayan number by 20? If it were purely base-20 the answer would be trivial: just stick a zero on the end (like multiplying any number by 10 in our system). But if you actually do that, you haven’t actually multiplied the entire number by 20. The second digit moves into the third digit which means it’s only been multiplied by 18, so the resulting number is a bit smaller than 20 times the number.

1)      Stick a zero on the end.

2)      Take what is now the third digit and multiply it by two (don’t change it in the actual number, just see what it is and multiply it by two in your head)

3)      Add that number to the second digit.

4)      Regroup to account for possible roll-over in the second digit (i.e. carry the one…or two as the case may be)

5)      There is no step 5 because you’ve already finished!

It really is that easy. Basically, what you’re doing is this: When you add a zero on the end every digit is indeed multiplied by 20, except for the second digit. Therefore, you need to add that difference back into the number to get the correct answer. This can result in the second digit being to large for its place (i.e. being 18 or over), so if you need to, you can subtract 18 one or two times (you won’t need to do it more than twice, I promise), and add the number of times you had to subtract 18 to the third digit. It’s like if you were to add 9 + 7. Well, you can’t cram “16” into a single digit, so you subtract 10 once, and add the one to next place.

Let’s do two examples:

A) Multiply 8.2.13 by 20

2)      Take the third digit and multiply by two (so that would be 4)

3)      Add that to the second digit: 8.2.17.0

4)      Regroup if necessary (since the second digit is less than 18, it doesn’t roll over)

Therefore 8.2.13 × 20 = 8.2.17.0

B) Multiply 7.5.12.16 by 20

2)  Take the third digit and multiply by two (so, 24)

3)  Add that to the second digit: 7.5.12.40.0

4)  Regroup if necessary (40 goes into 18 twice, with a remainder of 4, so we make the second digit 4 and add 2 to the third digit): 7.5.14.4.0

So anyway, that’s it. The only problem left is to see if the rolling-over ever stops. I mean, what if the number you’re dealing with keeps making higher digits continue to roll over all the way up? For example, if you have the number (in our system) 9999999999999 and add 1, the ones’ place becomes 0 and a 1 rolls over into the tens’ place, which also becomes zero and rolls over into the hundreds’ place…which also becomes zero and rolls over into the thousands’ place, and so on. It would be unfortunate if this were to happen in Mayan numerals, since all that writing would take time.

Fortunately, it doesn’t. Let’s take a Mayan number that would be like 999999etc: 19.19.17.19. If you were to add 1 to this number, the places would keep rolling over until you were left with 1.0.0.0.0. So, let’s see if that happens when you multiply by 20:

2)      Take the third place and multiply by two (17 × 2 = 34)

3)      Add that number into the second place digit (19 + 34 = 53): 19.19.17.53.0

4)      Regroup: 53 ÷ 18 = 2 remainder 17. So the third digit becomes 19 and the second becomes 17: 19.19.19.17.0

Phew! The rollover stops at the third place in all cases (I can say this since this is the case where the number are at their largest).

So that’s my solution. As my professor might say “it’s good clean fun!” So anyway…wait…hold on, what’s that at the end of the article?…

I have a program that can convert our number system into Mayan numbers? I don’t have a problem that can convert our number system into Mayan numbers. That’s means I need to make one, quick! My e-mail’s on the thing! And the deadline where everyone loses interest is in three days! ARRRRRGGGG!!!!!

See you on the 22nd!

Paper Update

Just a quick update, the paper I mentioned in my last post has been accepted for publication! I am incredibly excited as I’ve never had anything published before. The working title of the article is “The Mayan Number System and Calendar” and will be published in Mathematics Teacher this December (basically to coincide with the whole 2012 end of the world nonsense. Still it’s some sort of link) Of course, once it is published I will reveal the actual answer and everything.

Incredibly Awesome News

I’m probably going to have a paper published! On Mayan numeral systems of all things!

This past quarter, I took a class called “History of Mathematics”. It seemed interesting, there was nothing else to fill my schedule, and, as it turns out, it will could towards my elective credit for my degree. During the class we had a homework problem which asked to find a simple way of multiplying a Mayan number by 20. Nobody could solve it, and the teacher (Dr. Shirley Gray) stated that in all her years of teaching the class, no student of hers had ever solved the problem. She had even written the textbook’s authors and they didn’t have a solution either. Well, I solved it.

It seems like a simple problem. But to understand it, we must understand how Mayan numbers work. Mayan numbers are, essentially, a base-20 positional system. It is like ours, except that each successive digit it twenty times greater than the previous one (in ours, of course, each digit is only ten times greater).

Based on this, it should be trivial to multiply by 20: just add a zero on the end (like multiplying by 10 in our system). However, there is a slight problem. For some reason, the third digit in a Mayan number is only 18 times the previous digit (meaning that 1-0-0 in their system is equal to 360, not 400). Adding a zero on the end won’t result in a number twenty times greater.

So, how do you do it? Well, I can’t write the answer here because I don’t want to inadvertently mess up getting the paper published. Dr. Gray did grill me on whether I found the solution online (which I didn’t, of course. Plagiarism is a very serious academic offense and if I had found it online, I would have stated so in the first place. I understand her caution completely). What I will say, though, is that the solution is simple, quick, and you don’t even have to convert the Mayan number into our own system to do it (they didn’t use Hindu-Arabic numbers, naturally. It was system of lines and dots written vertically, with a strange shell shape for the number zero. See here).

Where the paper comes in is that Dr. Gray was already writing a paper on Mayan numbers and wants to include my finding in it. So it’s not really my paper, but it’s still a mention in some official publication. Which is more than I’ve ever done so far.

So, when the paper is published, I will announce it here, as well as the actual solution with the proof I developed. It will interesting to put on my resumé, right below ‘recording “Hollaback Girl” with Gwen Stefani’ (true story! Also, sorry…)

Happy τ Day!

That’s right, its τ Day! Wait, what?

Recently, I have found an interesting proposal regarding the number π (the ratio of the circumference of a circle to the diameter, or, 3.14159265358979…and so on…). It states that using π is not the best choice and that we should use a constant that is equal to 2π, which is 6.283185307179586…and so on… Basically, the ratio between the circumference of a circle to it radius is more basic a concept than circumference to diameter.

So what’s the difference? Well the main point is that in almost every single equation that you encounter in math and physics that comes with a π in it, it actually comes as 2π or some multiple thereof. Therefore, it is more convenient to simply write τ (called “tau”), one character, than 2π, which is two characters. Physicists like to simplify things as much as possible when they can.

Therefore τ Day is 6/28, which is today!

There are other reasons too. Conceptually, it is more intuitive. Travelling τ radians takes you once around a circle. Travelling τ/2 radians takes you halfway. One tenth of a circle is τ/10, and so on. For π, the values would be 2ππ, and π/5, which don’t really make sense. Travelling π/5 takes you a tenth of the way around a circle, rather than a fifth?

Anyway, I love it, and maybe I’ll try to use it on some test and see what happens (initially noting that τ = 2π, of course).